2007 IMS

Part of IMS

Subcontests

(8)

1

real number with a partition condition

x_{1},x_{2},\dots,x_{n}

are real number such that for each

i

\{x_{1},x_{2},\dots,x_{n}\}\backslash \{x_{i}\}

could be partitioned into two sets that sum of elements of first set is equal to the sum of the elements of the other. Prove that all of

x_{i}

's are zero. [hide="Hint"]It is a number theory problem.

1

Fixed point

T=\{(tq,1-t) \in\mathbb R^{2}| t \in [0,1],q\in\mathbb Q\}

Prove that each continuous function

f: T\longrightarrow T

has a fixed point.

1

Limit

\alpha,\beta

such that the following limit exists and is finite:

\lim_{x,y\rightarrow 0^{+}}\frac{x^{2\alpha}y^{2\beta}}{x^{2\alpha}+y^{3\beta}}

1

Unfair dices

Does there exist two unfair dices such that probability of their sum being

j

\left(\frac2{33},\frac4{33}\right)

2\leq j\leq 12

1

Commutative ring with 1

R

be a commutative ring with 1. Prove that

R[x]

has infinitely many maximal ideals.

1

Group with exactly n subgroups of index 2

Suppose there exists a group with exactly

n

subgroups of index 2. Prove that there exists a finite abelian group

G

that has exactly

n

subgroups of index 2.

1

A dense subset in R^2

\mathbb R^{2}

has a dense subset such that has no three collinear points.

1

A determinant Problem

Prove that: \det(A)=\frac{1}{n!}\left| \begin{array}{llllll}\mbox{tr}(A) & 1 & 0 & \ldots & \ldots & 0 \\ \mbox{tr}(A^{2}) & \mbox{tr}(A) & 2 & 0 & \ldots & 0 \\ \mbox{tr}(A^{3}) & \mbox{tr}(A^{2}) & \mbox{tr}(A) & 3 & & \vdots \\ \vdots & & & & & n-1 \\ \mbox{tr}(A^{n}) & \mbox{tr}(A^{n-1}) & \mbox{tr}(A^{n-2}) & \ldots & \ldots & \mbox{tr}(A) \end{array}\right|